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The hyperbolic random geometric graph $G_{\mathbb{H}}$ consists of a $N$ uniformly random points (within a disk of radius $R$ centred at the origin) of the hyperbolic plane $\mathbb{H}$, connected when within mutual hyperbolic distance $r$. Consider the native model, where nodes at $(r_1,\theta_1)$ and $(r_2,\theta_2)$ connect when $\operatorname{arcosh} \gamma < r$, where

$$ \gamma =\cosh (r_1) \cosh (r_2)-\cos (\delta ) \sinh (r_1) \sinh (r_2) $$ and $ \delta =\pi -| \pi -| \theta_1-\theta_2| |. $

This model has high clustering, and a power law degree distribution.

Is it possible to retain the power law, but reduce the clustering, by using a higher dimensional hyperbolic space like $\mathbb{H}^{3}$ or beyond, or letting the space get more negatively curved? This would lead to fewer triangles, but retain the tree-like structure of the graph.

This can help when trying to detach the power law from the clustering, without losing the geometry, for the purposes of studying dynamics on the graph.

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    $\begingroup$ What is a "uniformly random point" on the hyperbolic plane? $\endgroup$
    – R W
    Commented Nov 1, 2022 at 10:17
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    $\begingroup$ i.e. within a disk of radius $R$ centred at the origin of the hyperbolic plane. $\endgroup$
    – apg
    Commented Nov 1, 2022 at 10:29

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